In this paper, a boundary version of the Schwarz lemma is investigated. We consider a function f holomorphic in the unit disc D, f (a) = b, |a| < 1 and f (z) − α < α for z ∈ D, where α is a positive real number and 1 2 < α ≤ 1. We obtain sharp lower bounds on the angular derivative f (c) at the point c, where f (c) = 2α, |c| = 1.